Skip to main content

Advertisement

Springer Nature Link
Log in

Discrete Salp Swarm Algorithm for Euclidean Travelling Salesman Problem

  • Published:
Applied Intelligence Aims and scope Submit manuscript

Abstract

The Salp Swarm Algorithm (SSA) is one of the recently proposed swarm intelligence-based algorithms, which finds its motivation in the swarming behaviour of salps when navigating and foraging in the ocean. The SSA is essentially derived to solve optimization problems that have continuous search space. SSA is a simple to implement and competitive algorithm that has been proven helpful in a variety of real-world applications. It has been explored over several optimization problems so far. In this study, an enhanced discrete version of SSA is proposed to solve the Travelling Salesman Problem (TSP). As TSP is a combinatorial optimization problem, the classical SSA is modified using swap, shift and symmetry operators for global exploration and local exploitation. Also, the 2-opt method has been incorporated with the proposed algorithm to improve the local search ability of the algorithm while addressing discrete problems. The proposed Discrete Salp Swarm Algorithm (DSSA) has evaluated over 45 TSP instances, and the computational results showed that it is a promising algorithm. To assess its performance, the proposed algorithm’s results are compared with well-known algorithms such as Genetic Algorithm, Artificial Bee Colony, Spider Monkey Algorithm, Jaya Algorithm, Black Hole, Symbiotic Organism Search etc. The proposed algorithm significantly outperformed these algorithms for a majority of TSP instances.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13

Similar content being viewed by others

Explore related subjects

Discover the latest articles and news from researchers in related subjects, suggested using machine learning.

References

  1. Gavett JW, Plyter NV (1966) The optimal assignment of facilities to locations by branch and bound. Oper Res 14(2):210–232

    Google Scholar 

  2. Padberg M, Rinaldi G (1987) Optimization of a 532-city symmetric traveling salesman problem by branch and cut. Oper Res Lett 6(1):1–7

    MathSciNet  MATH  Google Scholar 

  3. Bellman R (1962) Dynamic programming treatment of the travelling salesman problem. Journal of the ACM (JACM) 9(1):61–63

    MathSciNet  MATH  Google Scholar 

  4. Gendreau M, Hertz A, Laporte G (1994) A tabu search heuristic for the vehicle routing problem. Manag Sci 40(10):1276–1290

    MATH  Google Scholar 

  5. Lawler EL (1985) The traveling salesman problem: a guided tour of combinatorial optimization, Wiley-Interscience Series in Discrete Mathematics

  6. Koopmans TC, Beckmann M (1957) Assignment problems and the location of economic activities

  7. Toth P, Vigo D (2002) The vehicle routing problem, SIAM

  8. de Werra D (1985) An introduction to timetabling. Eur J Oper Res 19(2):151–162

    MathSciNet  MATH  Google Scholar 

  9. Allahverdi A, Ng CT, Cheng TE, Kovalyov MY (2008) A survey of scheduling problems with setup times or costs. Eur J Oper Res 187(3):985–1032

    MathSciNet  MATH  Google Scholar 

  10. Papadimitriou CH, Steiglitz K (1998) Combinatorial optimization: algorithms and complexity, Courier Corporation

  11. Wong W (1995) Matrix representation and gradient flows for np-hard problems. J Optim Theory Appl 87(1):197–220

    MathSciNet  MATH  Google Scholar 

  12. Laporte G (1992) The traveling salesman problem: an overview of exact and approximate algorithms. Eur J Oper Res 59(2):231– 247

    MATH  Google Scholar 

  13. Laporte G, Nobert Y (1980) A cutting planes algorithm for the m-salesmen problem. J Oper Res Soc 31(11):1017–1023

    MathSciNet  MATH  Google Scholar 

  14. Dasgupta D, Michalewicz Z (1997) Evolutionary algorithms—an overview

  15. Slowik A, Kwasnicka H (2020) Evolutionary algorithms and their applications to engineering problems

  16. Mirjalili S, Gandomi AH, Mirjalili SZ, Saremi S, Faris H, Mirjalili SM (2017) Salp swarm algorithm: a bio-inspired optimizer for engineering design problems. Adv Eng Softw 114:163–191

    Google Scholar 

  17. Tubishat M, Idris N, Shuib L, Abushariah MA, Mirjalili S (2020) Improved salp swarm algorithm based on opposition based learning and novel local search algorithm for feature selection. Expert Syst Appl 145:113–122

    Google Scholar 

  18. Rizk-Allah RM, Hassanien AE, Elhoseny M, Gunasekaran M (2019) A new binary salp swarm algorithm: development and application for optimization tasks. Neural Comput & Applic 31(5):1641–1663

    Google Scholar 

  19. Zhang J, Wang Z, Luo X (2018) Parameter estimation for soil water retention curve using the salp swarm algorithm. Water 10(6):815–826

    Google Scholar 

  20. Ekinci S, Hekimoglu B (2018) Parameter optimization of power system stabilizer via salp swarm algorithm. In: 2018 5th international conference on electrical and electronic engineering (ICEEE), IEEE, pp 143–147

  21. Ibrahim HT, Mazher WJ, Ucan ON, Bayat O (2017) Feature selection using salp swarm algorithm for real biomedical datasets. IJCSNS 17(12):13–20

    Google Scholar 

  22. Tolba M, Rezk H, Diab AAZ, Al-Dhaifallah M (2018) A novel robust methodology based salp swarm algorithm for allocation and capacity of renewable distributed generators on distribution grids. Energies 11(10):25–56

    Google Scholar 

  23. Zhang H, Cai Z, Ye X, Wang M, Kuang F, Chen H, Li C, Li Y (2020) A multi-strategy enhanced salp swarm algorithm for global optimization

  24. Salgotra R, Singh U, Singh S, Singh G, Mittal N (2021) Self-adaptive salp swarm algorithm for engineering optimization problems. Appl Math Model 89:188–207

    MathSciNet  MATH  Google Scholar 

  25. Singh N, Singh S, Houssein EH (2020) Hybridizing salp swarm algorithm with particle swarm optimization algorithm for recent optimization functions

  26. Li E, Zhou J, Shi X, Jahed Armaghani D, Yu Z, Chen X, Huang P (2021) Developing a hybrid model of salp swarm algorithm-based support vector machine to predict the strength of fiber-reinforced cemented paste backfill. Engineering with Computers 37(4):3519–3540

    Google Scholar 

  27. Zhang J, Wang J. -S. (2020) Improved salp swarm algorithm based on levy flight and sine cosine operator. IEEE Access 8:99740–99771

    Google Scholar 

  28. Syed MA, Syed R (2019) Weighted salp swarm algorithm and its applications towards optimal sensor deployment, Journal of King Saud University-Computer and Information Sciences

  29. Neggaz N, Ewees AA, Abd Elaziz M, Mafarja M (2020) Boosting salp swarm algorithm by sine cosine algorithm and disrupt operator for feature selection. Expert Syst Appl 145:113–133

    Google Scholar 

  30. El-Ashmawi WH, Ali AF (2020) A modified salp swarm algorithm for task assignment problem. Appl Soft Comput 94:106–127

    Google Scholar 

  31. Panda N, Majhi SK (2020) Improved salp swarm algorithm with space transformation search for training neural network. Arab J Sci Eng 45(4):2743–2761

    Google Scholar 

  32. Meraihi Y, Ramdane-Cherif A, Mahseur M, Achelia D (2018) A chaotic binary salp swarm algorithm for solving the graph coloring problem. In: International Symposium on Modelling and Implementation of Complex Systems, Springer, pp 106–118

  33. Bairathi D, Gopalani D (2019) Salp swarm algorithm (ssa) for training feed-forward neural networks. In: Soft computing for problem solving, Springer, pp 521–534

  34. Verma S, Shiva CK (2020) A novel salp swarm algorithm for expansion planning with security constraints, Iranian Journal of Science and Technology

  35. El-Fergany AA, Hasanien HM (2020) Salp swarm optimizer to solve optimal power flow comprising voltage stability analysis. Neural Comput & Applic 32(9):5267–5283

    Google Scholar 

  36. Ibrahim RA, Ewees AA, Oliva D, Abd Elaziz M, Lu S (2019) Improved salp swarm algorithm based on particle swarm optimization for feature selection. Journal of Ambient Intelligence and Humanized Computing 10(8):3155–3169

    Google Scholar 

  37. Faris H, Mafarja MM, Heidari AA, Aljarah I, Ala’M A.-Z, Mirjalili S, Fujita H (2018) An efficient binary salp swarm algorithm with crossover scheme for feature selection problems. Knowl-Based Syst 154:43–67

    Google Scholar 

  38. Ateya AA, Muthanna A, Vybornova A, Algarni AD, Abuarqoub A, Koucheryavy Y, Koucheryavy A (2019) Chaotic salp swarm algorithm for sdn multi-controller networks. Engineering Science and Technology, an International Journal 22(4):1001–1012

    Google Scholar 

  39. Medjahed SA, Ouali M (2020) A new hybrid ssa-ta: Salp swarm algorithm with threshold accepting for band selection in hyperspectral images. Appl Soft Comput 95:106–113

    Google Scholar 

  40. Singh N, Son LH, Chiclana F, Magnot J.-P. (2020) A new fusion of salp swarm with sine cosine for optimization of non-linear functions. Engineering with Computers 36(1):185–212

    Google Scholar 

  41. Pintea C.-M., Pop PC, Chira C (2017) The generalized traveling salesman problem solved with ant algorithms. Complex Adaptive Systems Modeling 5(1):1–9

    Google Scholar 

  42. Xiao Y, Jiao J, Pei J, Zhou K, Yang X (2018) A multi-strategy improved ant colony algorithm for solving traveling salesman problem. In: IOP Conference Series: Materials Science and Engineering, Vol. 394, IOP Publishing, p 042101

  43. Gharehchopogh FS, Abdollahzadeh B (2021) An efficient harris hawk optimization algorithm for solving the travelling salesman problem

  44. Ali IM, Essam D, Kasmarik K (2020) A novel design of differential evolution for solving discrete traveling salesman problems. Swarm and Evolutionary Computation 52:100607

    Google Scholar 

  45. Benyamin A, Farhad SG, Saeid B (2021) Discrete farmland fertility optimization algorithm with metropolis acceptance criterion for traveling salesman problems. Int J Intell Syst 36(3):1270–1303

    Google Scholar 

  46. Boryczka U, Szwarc K (2019) The harmony search algorithm with additional improvement of harmony memory for asymmetric traveling salesman problem. Expert Syst Appl 122:43–53

    Google Scholar 

  47. Zhou Y, Wang R, Zhao C, Luo Q, Metwally MA (2019) Discrete greedy flower pollination algorithm for spherical traveling salesman problem. Neural Comput & Applic 31(7):2155–2170

    Google Scholar 

  48. Bi J, Zhou G, Zhou Y, Luo Q, Deng W (2022) Artificial electric field algorithm with greedy state transition strategy for spherical multiple traveling salesmen problem. International Journal of Computational Intelligence Systems 15(1):1–24

    Google Scholar 

  49. Papadimitriou CH (1977) The euclidean travelling salesman problem is np-complete. Theor Comput Sci 4(3):237–244

    MATH  Google Scholar 

  50. Croes GA (1958) A method for solving traveling-salesman problems. Oper Res 6(6):791–812

    MathSciNet  MATH  Google Scholar 

  51. Tarantilis CD, Kiranoudis CT (2007) A flexible adaptive memory-based algorithm for real-life transportation operations: Two case studies from dairy and construction sector. Eur J Oper Res 179(3):806–822

    MATH  Google Scholar 

  52. Bianchessi N, Righini G (2007) Heuristic algorithms for the vehicle routing problem with simultaneous pick-up and delivery. Comput Oper Res 34(2):578–594

    MATH  Google Scholar 

  53. Cinar AC, Korkmaz S, Kiran MS (2020) A discrete tree-seed algorithm for solving symmetric traveling salesman problem. Engineering Science and Technology, an International Journal 23(4):879–890

    Google Scholar 

  54. Zhou X, Gao DY, Yang C, Gui W (2016) Discrete state transition algorithm for unconstrained integer optimization problems. Neurocomputing 173:864–874

    Google Scholar 

  55. Gunduz M, Aslan M (2021) Djaya: a discrete jaya algorithm for solving traveling salesman problem. Appl Soft Comput 105:107–122

    Google Scholar 

  56. Chunhua Y, Xiaolin T, Xiaojun Z, Weihua G (2012) State transition algorithm for traveling salesman problem. In: Proceedings of the 31st Chinese Control Conference, IEEE, pp 2481–2485

  57. Gündüz M., Kiran MS, Özceylan E. (2015) A hierarchic approach based on swarm intelligence to solve the traveling salesman problem. Turk J Electr Eng Comput Sci 23(1):103–117

    Google Scholar 

  58. Hatamlou A (2018) Solving travelling salesman problem using black hole algorithm. Soft Comput 22(24):8167–8175

    Google Scholar 

  59. Khan I, Maiti MK (2019) A swap sequence based artificial bee colony algorithm for traveling salesman problem. Swarm and Evolutionary Computation 44:428–438

    Google Scholar 

  60. Akhand M, Ayon SI, Shahriyar S, Siddique N, Adeli H (2020) Discrete spider monkey optimization for travelling salesman problem. Appl Soft Comput 86:105–121

    Google Scholar 

  61. Ezugwu A. E. -S., Adewumi AO (2017) Discrete symbiotic organisms search algorithm for travelling salesman problem. Expert Syst Appl 87:70–78

    Google Scholar 

  62. Mahi M, Baykan ÖK, Kodaz H (2015) A new hybrid method based on particle swarm optimization, ant colony optimization and 3-opt algorithms for traveling salesman problem. Appl Soft Comput 30:484–490

    Google Scholar 

  63. Zhou Y, Luo Q, Chen H, He A, Wu J (2015) A discrete invasive weed optimization algorithm for solving traveling salesman problem. Neurocomputing 151:1227–1236

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, India

    Karuna Panwar & Kusum Deep

Authors
  1. Karuna Panwar
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Kusum Deep
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Karuna Panwar.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Panwar, K., Deep, K. Discrete Salp Swarm Algorithm for Euclidean Travelling Salesman Problem. Appl Intell 53, 11420–11438 (2023). https://doi.org/10.1007/s10489-022-03976-5

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10489-022-03976-5

Keywords